The Topological Properties Of Fractals With Positive Lebesgue Measure

The research on the measure and topological properties of fractal sets is one of the key topics that people are concerned about.As an important class of typical fractal sets,self-similar sets have a relatively simple structure,which has been widely studied.Peres and Solomyak[50]proposed the following question:assume that a self-similar set in R^d has positive d-dimensional Lebesgue measure,must it have nonempty interior?Schief[53]gives a partial answer to the above question.He proves that a self-similar set with posutive Lebesgue measure must have nonempty interior under the open set condition.Subsequently,Zerner[59]and Peres et al.[49]extend Schief's result to self-similar sets with the weak separation condition and self-conformal sets with the open set condition.In this thesis,we mainly study the following problem:What structure ensures the equivalence of positive Lebesgue measure and nonempty interior for fractals?In Chapter 3,based on the notion of BPI spaces proposed by David and Semmes[11],we introduce BBI spaces.Roughly speaking,for any pair of balls B₁and B₂in the space,we can find a relatively large ball in B₁and a subset in B₂such that they look approximately the same in terms of conformal bilipschitz equivalence.Futhermore,we prove that the“self-similar”structure described by BBI spaces ensures the equivalence of positive Lebesgue measure and nonempty interior.In Chapter 4,we apply the result in Chapter 3 to self-conformal sets that have bounded distortion property and satisfy the weak separation condition,which generalizes Schief's,Zerner's and Pere's corresponding result.In Chapter 5,based on the results of Schief,we consider a class of generalized self-similar sets and obtain the following results:Let{S_i}_i?1be a sequence of iterated function system of similitudes on R^d,and E the generalized self-similar set generated by it.Suppose that there exists a nonempty open set U?R^dsuch that each S_isatisfies the open set condition with respect to U,and that,the similarity dimension of S_i is equal to the space dimension d,then E=U.Moreover,we give some examples.